# Topological property

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are *not* homeomorphic, it is sufficient to find a topological property which is not shared by them.

## Common topological properties

### Cardinal functions

- The cardinality |
*X*| of the space*X*. - The cardinality
*τ*(*X*) of the topology of the space*X*. *Weight**w*(*X*), the least cardinality of a basis of the topology of the space*X*.*Density**d*(*X*), the least cardinality of a subset of*X*whose closure is*X*.

### Separation

Note that some of these terms are defined differently in older mathematical literature; see history of the separation axioms.

**T**or_{0}**Kolmogorov**. A space is Kolmogorov if for every pair of distinct points*x*and*y*in the space, there is at least either an open set containing*x*but not*y*, or an open set containing*y*but not*x*.**T**or_{1}**Fréchet**. A space is Fréchet if for every pair of distinct points*x*and*y*in the space, there is an open set containing*x*but not*y*. (Compare with T_{0}; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T_{1}if all its singletons are closed. T_{1}spaces are always T_{0}.**Sober**. A space is sober if every irreducible closed set*C*has a unique generic point*p*. In other words, if*C*is not the (possibly nondisjoint) union of two smaller closed subsets, then there is a*p*such that the closure of {*p*} equals*C*, and*p*is the only point with this property.**T**or_{2}**Hausdorff**. A space is Hausdorff if every two distinct points have disjoint neighbourhoods. T_{2}spaces are always T_{1}.**T**or_{2½}**Urysohn**. A space is Urysohn if every two distinct points have disjoint*closed*neighbourhoods. T_{2½}spaces are always T_{2}.**Completely T**or_{2}**completely Hausdorff**. A space is completely T_{2}if every two distinct points are separated by a function. Every completely Hausdorff space is Urysohn.**Regular**. A space is regular if whenever*C*is a closed set and*p*is a point not in*C*, then*C*and*p*have disjoint neighbourhoods.**T**or_{3}**Regular Hausdorff**. A space is regular Hausdorff if it is a regular T_{0}space. (A regular space is Hausdorff if and only if it is T_{0}, so the terminology is consistent.)**Completely regular**. A space is completely regular if whenever*C*is a closed set and*p*is a point not in*C*, then*C*and {*p*} are separated by a function.**T**,_{3½}**Tychonoff**,**Completely regular Hausdorff**or**Completely T**. A Tychonoff space is a completely regular T_{3}_{0}space. (A completely regular space is Hausdorff if and only if it is T_{0}, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff.**Normal**. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity.**T**or_{4}**Normal Hausdorff**. A normal space is Hausdorff if and only if it is T_{1}. Normal Hausdorff spaces are always Tychonoff.**Completely normal**. A space is completely normal if any two separated sets have disjoint neighbourhoods.**T**or_{5}**Completely normal Hausdorff**. A completely normal space is Hausdorff if and only if it is T_{1}. Completely normal Hausdorff spaces are always normal Hausdorff.**Perfectly normal**. A space is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal.**T**or_{6}**Perfectly normal Hausdorff**, or**perfectly T**. A space is perfectly normal Hausdorff, if it is both perfectly normal and T_{4}_{1}. A perfectly normal Hausdorff space must also be completely normal Hausdorff.**Discrete space**. A space is discrete if all of its points are completely isolated, i.e. if any subset is open.

### Countability conditions

**Separable**. A space is separable if it has a countable dense subset.**First-countable**. A space is first-countable if every point has a countable local base.**Second-countable**. A space is second-countable if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf.

### Connectedness

**Connected**. A space is connected if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the empty set and itself.**Locally connected**. A space is locally connected if every point has a local base consisting of connected sets.**Totally disconnected**. A space is totally disconnected if it has no connected subset with more than one point.**Path-connected**. A space*X*is path-connected if for every two points*x*,*y*in*X*, there is a path*p*from*x*to*y*, i.e., a continuous map*p*: [0,1] →*X*with*p*(0) =*x*and*p*(1) =*y*. Path-connected spaces are always connected.**Locally path-connected**. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected.**Simply connected**. A space*X*is simply connected if it is path-connected and every continuous map*f*: S^{1}→*X*is homotopic to a constant map.**Locally simply connected**. A space*X*is locally simply connected if every point*x*in*X*has a local base of neighborhoods*U*that is simply connected.**Semi-locally simply connected**. A space*X*is semi-locally simply connected if every point has a local base of neighborhoods*U*such that*every*loop in*U*is contractible in*X*. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a universal cover.**Contractible**. A space*X*is contractible if the identity map on*X*is homotopic to a constant map. Contractible spaces are always simply connected.**Hyper-connected**. A space is hyper-connected if no two non-empty open sets are disjoint. Every hyper-connected space is connected.**Ultra-connected**. A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected.**Indiscrete**or**trivial**. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have the trivial topology.

### Compactness

**Compact**. A space is compact if every open cover has a finite*subcover*. Some authors call these spaces**quasicompact**and reserve compact for Hausdorff spaces where every open cover has finite subcover. Compact spaces are always Lindelöf and paracompact. Compact Hausdorff spaces are therefore normal.**Sequentially compact**. A space is sequentially compact if every sequence has a convergent subsequence.**Countably compact**. A space is countably compact if every countable open cover has a finite subcover.**Pseudocompact**. A space is pseudocompact if every continuous real-valued function on the space is bounded.**σ-compact**. A space is σ-compact if it is the union of countably many compact subsets.**Lindelöf**. A space is Lindelöf if every open cover has a countable subcover.**Paracompact**. A space is paracompact if every open cover has an open locally finite refinement. Paracompact Hausdorff spaces are normal.**Locally compact**. A space is locally compact if every point has a local base consisting of compact neighbourhoods. Slightly different definitions are also used. Locally compact Hausdorff spaces are always Tychonoff.**Ultraconnected compact**. In an ultra-connected compact space*X*every open cover must contain*X*itself. Non-empty ultra-connected compact spaces have a largest proper open subset called a**monolith**.

### Metrizability

**Metrizable**. A space is metrizable if it is homeomorphic to a metric space. Metrizable spaces are always Hausdorff and paracompact (and hence normal and Tychonoff), and first-countable. Moreover a topological space (X,T) is said to be metrizable if there exists a metric for X such that the metric topology T(d) is identical with the topology T.**Polish**. A space is called Polish if it is metrizable with a separable and complete metric.**Locally metrizable**. A space is locally metrizable if every point has a metrizable neighbourhood.

### Miscellaneous

**Baire space**. A space*X*is a Baire space if it is not meagre in itself. Equivalently,*X*is a Baire space if the intersection of countably many dense open sets is dense.**Topological Homogeneity**. A space*X*is (topologically) homogeneous if for every*x*and*y*in*X*there is a homeomorphism*f*:*X*→*X*such that*f*(*x*) =*y*. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous.**Finitely generated**or**Alexandrov**. A space*X*is Alexandrov if arbitrary intersections of open sets in*X*are open, or equivalently if arbitrary unions of closed sets are closed. These are precisely the finitely generated members of the category of topological spaces and continuous maps.**Zero-dimensional**. A space is zero-dimensional if it has a base of clopen sets. These are precisely the spaces with a small inductive dimension of*0*.**Almost discrete**. A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.**Boolean**. A space is Boolean if it is zero-dimensional, compact and Hausdorff (equivalently, totally disconnected, compact and Hausdorff). These are precisely the spaces that are homeomorphic to the Stone spaces of Boolean algebras.**Reidemeister torsion****-resolvable**. A space is said to be κ-resolvable[1] (respectively: almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (respectively: almost disjoint over the ideal of nowhere dense subsets). If the space is not -resolvable then it is called -irresolvable.**Maximally resolvable**. Space is maximally resolvable if it is -resolvable, where . Number is called dispersion character of .**Strongly discrete**. Set is strongly discrete subset of the space if the points in may be separated by pairwise disjoint neighborhoods. Space is said to be strongly discrete if every non-isolated point of is the accumulation point of some strongly discrete set.

## See also

## References

- Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán (2008). "Resolvability and monotone normality".
*Israel Journal of Mathematics*.**166**(1): 1–16. arXiv:math/0609092. doi:10.1007/s11856-008-1017-y. ISSN 0021-2172.

## Bibliography

- Willard, Stephen (1970).
*General topology*. Reading, Mass.: Addison-Wesley Pub. Co. p. 369. ISBN 9780486434797.